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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Using flows to construct Hilbert space factors of function spaces


Author: James Keesling
Journal: Trans. Amer. Math. Soc. 161 (1971), 1-24
MSC: Primary 54.28; Secondary 57.00
DOI: https://doi.org/10.1090/S0002-9947-1971-0283751-8
MathSciNet review: 0283751
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Abstract: Let X and Y be metric spaces. Let $ G(X)$ be the group of homeomorphisms of X with the compact open topology. The main result of this paper is that if X admits a nontrivial flow, then $ G(X)$ is homeomorphic to $ G(X) \times {l_2}$ where $ {l_2}$ is separable infinite-dimensional Hilbert space. The techniques are applied to other function spaces with the same result. Two such spaces for which our techniques apply are the space of imbeddings of X into Y, $ E(X,Y)$, and the space of light open mappings of X into (or onto) Y, LO (X, Y). Some applications of these results are given. The paper also uses flows to show that if X is the $ \sin (1/x)$-curve, then $ G(X)$ is homeomorphic to $ {l_2} \times N$, where N is the integers.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0283751-8
Keywords: Flow, group of homeomorphisms, space of imbeddings, space of light mappings, infinite-dimensional manifold, Hilbert space, Hilbert cube, hyperspace, negligible subsets
Article copyright: © Copyright 1971 American Mathematical Society